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1
∫
∫
Ω
Ω
therefore
0
1 2
2 2
0
1 1 2
d
d
d = d
d
d
∫ ∫∫ d 




G
x
x R mx x
x
i x
x
γ Ω γi
The integral on the right-hand side can be transformed to read
Ω2 2
0
1
0
2
d
d
R mx d d
x
x x
x
i ∫ ∫ 


γ
and eqn 7.19 can finally be expressed as
ω ω α i i
2
nr
= 2 + Ω2 (7.20)
where ωnr is the natural frequency of the non-rotating blade and
α
γ
γ
i
x
i
i
mx x x x
m x
=
(d /d ) d d
d
0
1
0
2
0
1
2
∫ ∫
∫
(7.21)
Equation 7.20 is Southwell’s formula.
To consider a specific case, let us use Southwell’s formula to calculate the first
bending frequency of a uniform blade hinged at the root (k2 = 0.004) having the mode
shape found earlier, namely
S2(x) = –3.15x + 4.15x3(10/3 – 10x/3 + x2)
Substituting in eqn 7.21 with m constant we find
α2 = 8.241
α1 being unity and corresponding to the rigid blade mode. The frequency of the nonrotating
blade is obtained from the standard results4 for a pinned-free non-rotating
uniform beam, which give
Structural dynamics of elastic blades 249
ω μ
μ
nr
2 4
4 4 2 2
= /
=
EI m
R k Ω
where μ is a constant given in the above reference.
If we take Ω = 25 rad/s, then k2Ω2 = 2.5 and the value of (μR)4 is 237.7.
Hence
ω nr
2 = 594.3
and Southwell’s formula for this case is therefore
ω 2
2 = 594.3 + 8.241Ω2
The frequency at Ω = 25 rad/s is 3.031Ω, which is higher than the value we
calculated before but this is to be attributed to the fact that the assumed mode shape,
which was based on only two functions, differs considerably from the more exact
value, and the departure from the true shape is equivalent to the imposition of constraints
which effectively increase the stiffness.
Southwell’s formula is very useful for showing the effect of the rotor speed on the
natural frequency of the blade. Strictly speaking, α
i is not constant because the mode
shape would be expected to change slightly with Ω. However, assuming α
i constant,
we see from eqn 7.20 that as the rotational speed becomes very large we have
ωi
→ Ω√α
i
The curves of eqn 7.20 for the various modes are often plotted in conjunction with a
‘spoke’ diagram, Fig. 7.7. One can see at a glance if there are any natural frequencies
of the blade which coincide with a harmonic of the rotor speed, indicating the possibility
of resonance.
(c) The Galerkin method
We start with the blade bending equation
ωi
Ω√α3
5Ω
4Ω
3Ω
2Ω
1Ω
Ω√α2
Rigid-blade mode
2nd bending
mode
1st
bending
mode
Ω
Fig. 7.7 ‘Spoke’ diagram of bending frequencies
250 Bramwell’s Helicopter Dynamics
d
d
d
d
– d
d
d
d
– = 0
2
2
2
2
2 22 4
x
EI
S
x
R
x
G
S
x
n n m RS
n n

 

 


λ Ω (7.11)
and suppose as before, that
Sn(x) = A1γ1(x) + A2γ2(x) + … + Aiγi(x) + … + Anγn(x) (7.22)
We now substitute for Sn(x) in eqn 7.11, multiply each term by γ
i, integrate from
0 to 1, and equate the result to zero. When i is made to take the successive values
1, 2, …, n, we obtain n simultaneous equations which determine the coefficients Ai.
Duncan has shown5 that this process is equivalent to finding the error ε (x) which
results from substituting the approximate solution eqn 7.22 into eqn 7.11 and finding
the stationary values of the integral
J dx
0
1
≡ ∫ ε 2
by satisfying the condition
∂J/∂Ai = 0
　
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